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\markboth{\hfil Pullback permanence \hfil EJDE--2002/72}
{EJDE--2002/72\hfil Jose A. Langa \& Antonio Su\'arez \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 72, pp. 1--20. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
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Pullback permanence for non-autonomous partial differential equations
%
\thanks{ {\em Mathematics Subject Classifications:}
35B05, 35B22, 35B41, 37L05.
\hfil\break\indent
{\em Key words:} Non-autonomous differential equations,
pullback attractors, \hfil\break\indent
comparison techniques, permanence.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted May 14, 2001. Published August 8, 2002.
\hfil\break\indent
Partly supported by projects DGICYT PB98-1134 and CICYT MAR98-0486.} }
\date{}
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\author{Jose A. Langa \& Antonio Su\'arez}
\maketitle
\begin{abstract}
A system of differential equations is permanent if there
exists a fixed bounded set of positive states strictly
bounded away from zero to which, from a time on, any positive
initial data enter and remain. However, this fact does not happen
for a differential equation with general non-autonomous terms.
In this work we introduce the concept of pullback permanence,
defined as the existence of a time dependent set of positive
states to which all solutions enter and remain for suitable
initial time. We show this behaviour in the non-autonomous logistic
equation $u_{t}-\Delta u=\lambda u-b(t)u^{3}$, with $b(t)>0$ for all
$t\in \mathbb{R}$, $\lim_{t\to \infty }b(t)=0$.
Moreover, a bifurcation scenario for the asymptotic behaviour of
the equation is described in a neighbourhood of the first eigenvalue
of the Laplacian. We claim that pullback permanence can be a suitable
tool for the study of the asymptotic dynamics for general
non-autonomous partial differential equations.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\section{Introduction}
One of the main questions for a mathematical model from a natural
phenomena is that of the long-time behaviour of its solutions. In
particular, of special interest for an ecological model is to
predict the long time persistence of the species being modelled.
In this sense, we can look for strictly positive globally
attracting equilibria (or stationary solutions) for the
corresponding partial differential equation associated to the
model. But only in a restricted set of systems we can assure the
existence of stationary solutions. However, the concept of
\textit{global attractor} serves to put some light in the
understanding of the asymptotic behaviour of many dissipative
systems (Hale \cite{Hale}, Temam \cite{Temam}). Indeed, we can
infer \textit{uniform persistence or permanence} (Cantrell et al.
\cite{CantrellCosner}, \cite{CantrellCosnerHu}) of solutions from
the presence of a globally attracting positive set rather that a
single attracting equilibrium. A system is said to have uniform
persistence (Butler et al. \cite{Butler} ) if there exists a
positive set which is bounded away from zero and globally
attracting for all positive solutions. Note that this allows the
systems to have a more complex dynamics, so that a wider set of
(more natural) situations can be considered. On the other hand, we
lose some information on the location and size of these new sets,
so that any study of their \textit{structure} would be useful.
There exists a substantial literature on this subject for
autonomous differential equations (Hale and Waltman
\cite{HaleWaltman}, Hutson and Schmitt \cite{HutsonScmith}). The
system is said to be \textit{permanent} if it is also dissipative,
i.e., the orbits enter into a bounded set in a finite time.
Afterwards, Cao and Gard \cite{CaoGard} introduced the concept of
\textit{practical persistence}, defined as uniform persistence
together with some information on the location of the positive
attractor.
In this work we study problems on the permanence of positive
solutions for the following non-autonomous logistic equation
\begin{equation*}
u_{t}-\Delta u=\lambda u-b(t)u^{3},
\end{equation*}
with $b(t)>0$, the interesting case being when we impose
$\lim_{t\to +\infty }b(t)=0$. Previous works for
non-autonomous equations focus on the periodic or bounded by
periodic functions in time cases (Cantrell and Cosner
\cite{CantrellCosner}, Burton and Hutson \cite{Burton}, see also
Nkashama \cite{Nkas} for the finite dimensional case with bounded
and strictly positive non-autonomous terms). We treat a more
general case, in that we allow the equation a very weak
dissipation effect as time goes to infinite, and so previous works
in the literature are not valid for our purposes.
The situation can be summarized as follows: when the parameter
$\lambda <\lambda _{1}$, with $\lambda _{1}$ the first eigenvalue
of the negative Laplacian, we get the existence of the zero
solution as a globally attracting set. However, a drastic change
in the asymptotic behaviour happens as the parameter $\lambda $
crosses the value $\lambda _{1}$. We describe in some detail this
bifurcation scenario (Sections 3 and 4). Indeed, we firstly show
that the equation leads to an order-preserving system and the
method of sub and super solutions (Pao \cite{Pao}, Hess \cite
{Hess}) can be adapted to this case. Afterwards, we find a non
bounded order interval depending on time in which all the
asymptotic behaviour forward in time takes place (Section 4). That
is, there does not exist any bounded absorbing set for the
problem, and so no result on permanence in the sense of Cantrell
and Cosner \cite{CantrellCosner} can be expected.
However, very recently the theory of attractors for general
non-autonomous differential equations has been introduced (Cheban
et al. \cite{cheban}, Kloeden and Schamalfuss
\cite{KloedSchmalfuss}; see also Crauel and Flandoli
\cite{CrauelFland}, Crauel et al. \cite{CrauelFlanDe}, for the
same concept in a stochastic framework). In this case, the
semigroup becomes a process, that is, a two-time dependent
operator (Sell \cite{Sell}), where the dependence on initial time
is as important as that on the final time. When the non-autonomous
terms are periodic or quasi-periodic, the same concept of
attractor in Temam \cite{Temam} or Hale \cite{Hale} can be used
for these situations (Sell \cite{Sell}, Chepyzhov and Vishik
\cite{ChepVishik94}). But important changes in the concept must be
introduced when we deal with general non-autonomous terms.
Chepyzhov and Vishik \cite{ChepVishik94} define \textit{kernel}
and \textit{kernel sections. }This last concept is similar to that
defined in Cheban et al. \cite{cheban} as \textit{cocycle or
pullback attractor}. In our opinion, this is one of the right
concepts to define the attractor for a general non-autonomous
differential equation, as some results on the upper-semicontinuity
of pullback attractors to the (autonomous) global attractor show
(Caraballo and Langa \cite{CaraballoLa}). The attractor in this
situation is a time-dependent family of compact sets, invariant
with respect to the cocycle and attracting \ from `$-\infty $'
(see Definition~\ref{atrac}).
We apply the theory of pullback attractors to our non-autonomous logistic
equation. We also apply a result on the upper semicontinuity of this
pullback attractor to the global attractor for the autonomous equation. This
reinforces the choice of working with the pullback attractor to study the
asymptotic behaviour of non-autonomous equations.
While forward in time we have not information on the stability of the
equation when $\lambda >\lambda _{1}$, we describe a bifurcation scenario at
the parameter value $\lambda =\lambda _{1}$ from the pullback procedure: the
zero solution becomes unstable for $\lambda >\lambda _{1}$ and there exists
a transfer of stability to the pullback attractor, which is a set strictly
bigger than the zero solution, so that a result on permanence follows. We
think this is the sensible concept for permanence for general non-autonomous
partial differential equations (Definition \ref{permanence}). In Section 4.4
we are able to give more information on the structure of this pullback
attractor and so on the bifurcation phenomena. Indeed, by introducing the
concepts of sub-trajectories, super-trajectories and complete trajectories
for non-autonomous systems in Section 3, as generalization of the theory of
sub and super-equilibria in the sense of Hess \cite{Hess}, Arnold and
Chueshov \cite{ArnoldChu} and Chueshov \cite{Chues}, we describe the
existence of a maximal complete trajectory on the attractor with some
stability properties. We give a general theorem which can be applied to more
general situations.
Finally, some conclusions and possible generalizations are given in the
final Section.
\section{Non-autonomous attractors}
In this section, we introduce the general framework in which the theory of
attractors for non-autonomous systems is going to be studied (see Cheban et
al. \cite{cheban} and Schmalfuss \cite{Schmalfuss2}). In a first step, we
define processes as two-time dependent operators related with the solutions
of non-autonomous differential equations. In this way, we are able to treat
these equations as dynamical systems. Secondly, we write the general
definitions of invariance, absorption and attraction and we finish with a
general theorem on the existence of global attractors for these kind of
equations. Finally, we give the definition of permanence for non-autonomous
partial differential equations.
Let $(X,d)$ be a complete metric space (with the metric $d)$ with an order
relation `$\leq $' and $\{S(t,s)\}_{t\geq s}$, $t,s\in \mathbb{R}$ be a
family of mappings satisfying:
\begin{description}
\item[i)] $S(t,s)S(s,\tau )u=S(t,\tau )u$, for all $\tau \leq s\leq t$,
$u\in X$
\item[ii)] $u\mapsto S(t,\tau )u$ is continuous in $X$.
\end{description}
This map is called a process. In general, we have to consider $S(t,\tau )u$
as the solution of a non-autonomous equation at time $t$ with initial
condition $u$ at time$\ \tau $.
Let $\mathcal{D}$ be a non-empty set of parameterized families of
non-empty bounded sets $\left\{ D\left( t\right) \right\} _{t\in
\mathbb{R}}$. In particular, we could have $D\left( t\right)
\equiv B\in \mathcal{D}$, where $B\subset X$ is a bounded set. In
what follows, we will consider fixed this \textit{base of
attraction} $\mathcal{D}$, so that the concepts of absorption and
attraction in our analysis are always referred to it.
For $A,B\subset X$ define the Hausdorff semidistances as:
\begin{equation*}
\mathop{\rm dist}(A,B)=\sup_{a\in A}\inf_{b\in B}d(a,b),\quad
\mathop{\rm Dist}(A,B)=\inf_{a\in A}\inf_{b\in B}d(a,b).
\end{equation*}
\begin{definition} \rm
Given $t_{0}\in \mathbb{R}$, we say that a bounded set
$K(t_{0})\subset X$ is attracting
at time $t_{0}$ if for every $\left\{ D\left( t\right) \right\} \in \mathcal{%
D}$ we have that
\begin{equation*}
\lim_{\tau \to -\infty }\mathop{\rm dist}(S(t_{0},\tau )D\left( \tau \right)
,K(t_{0}))=0.
\end{equation*}
A family $\{K(t)\}_{t\in \mathbb{R}}$ is attracting if $K(t_{0})$ is
attracting at time $t_{0}$, for all $t_{0}\in \mathbb{R}$.
\end{definition}
The previous concept considers a fixed final time and moves the initial time
to $-\infty $. Note that this does not mean that we are going backwards in
time, but we consider the state of the system at time $t_{0}$ starting at $%
\tau \to -\infty $. This is called \textit{pullback attraction} in
the literature (cf. \cite{KloedSchmalfuss}, \cite{Schmalfuss2}).
\begin{definition}\label{defi2} \rm
Given $t_{0}\in \mathbb{R}$, we say that a bounded
set $B(t_{0})\subset X$ is absorbing at time $t_{0}$ if for every
$\left\{ D\left( t\right) \right\} \in \mathcal{D}$ there exists
$T=T(t_0,D)\in \mathbb{R}$ such that
\begin{equation*}
S(t_{0},\tau )D\left( \tau \right) \subset B(t_{0}),\text{ for all }\tau
\leq T.
\end{equation*}
A family $\{B(t)\}_{t\in \mathbb{R}}$ is absorbing if $B(t_{0})$ is
absorbing at time $t_{0}$, for all $t_{0}\in \mathbb{R}$.
\end{definition}
Note that every absorbing set at time $t_{0}$ is attracting.
\begin{definition} \rm
Let $\{B(t)\}_{t\in \mathbb{R}}$ be a family of subsets of $X$. This family
is said to be invariant with respect to the process $S$ if
\begin{equation*}
S(t,\tau )B(\tau )=B(t),\text{ for all }\left( \tau ,t\right) \in \mathbb{R}%
^{2},\tau \leq t.
\end{equation*}
\end{definition}
Note that this property is a generalization of the classical property of
invariance for semigroups. However, in this case we have to define the
invariance with respect to a family of sets depending on a parameter.
We define the \textit{omega-limit set} at time $t_{0}$ of $D\equiv \left\{
D\left( t\right) \right\} \in \mathcal{D}$ as
\begin{equation*}
\Lambda (D,t_{0})=\cap _{s\leq t_{0}}\overline{\cup _{\tau \leq
s}S(t_{0},\tau )D(\tau )}.
\end{equation*}
From now on, we assume that there exists a family $\{K(t)\}_{t\in \mathbb{R}%
} $ of compact absorbing sets, that is, $K(t)\subset X$ is non-empty,
compact and absorbing for each $t\in \mathbb{R}$. Note that, in this case, $%
\Lambda (D,t_{0})\subset K(t_{0})$, for all $\left\{ D\left( t\right)
\right\} \in \mathcal{D}$, $t_{0}\in \mathbb{R}$. As in the autonomous case,
it is not difficult to prove that under these conditions $\Lambda (D,t_{0})$
is non-empty, compact and attracts $\left\{ D\left( t\right) \right\} \in
\mathcal{D}$ at time $t_{0}$. The proof is similar to that in Crauel et al.
\cite{CrauelFlanDe}, where the set $\mathcal{D}$ consists only of bounded
sets.
\begin{definition}\label{atrac} \rm
The family of compact sets $\{ {\cal A} (t)\}_{t\in \mathbb{R}}$
is said to be the global non-autonomous (or
pullback) attractor associated to the process $S$ if it is invariant,
attracting every $\left\{ D\left( t\right) \right\} \in \mathcal{D}$ (for
all $t_{0}\in \mathbb{R}$) and minimal in the sense that if $\{C(t)\}_{t\in
\mathbb{R}}$ is another family of closed attracting sets, then
${\cal A}(t)\subset C(t)$ for all $t\in \mathbb{R}$.
\end{definition}
\begin{remark}\rm
Chepyzhov and Vishik \cite{ChepVishik93} define the concept of kernel
sections for non-autonomous dynamical systems which corresponds to our
definition of global non-autonomous attractor with $\{D(t)\}\equiv B\subset
X $ bounded.
\end{remark}
The general result on the existence of non-autonomous attractors is a
generalization of the abstract theory for autonomous dynamical systems
(Temam \cite{Temam}, Hale \cite{Hale}):
\begin{theorem}
[Crauel et al. \cite{CrauelFlanDe}, Schmalfuss \cite{Schmalfuss2}]
Assume that there exists a family of compact absorbing sets.
Then the family $\{{\cal A}(t)\}_{t\in \mathbb{R}}$ defined by
\begin{equation*}
{\cal A}(t)=\overline{\cup _{D\in \mathcal{D}}\Lambda (D,t)}
\end{equation*}
is the global non-autonomous attractor.
\end{theorem}
\begin{remark} \rm
All the general theory of non-autonomous attractors can be written
in the framework of cocycles (cf., among others, Cheban et al.
\cite{cheban}, Crauel and Flandoli \cite{CrauelFland}, Kloeden and
Schmalfuss \cite {KloedSchmalfuss}, Schmalfuss
\cite{Schmalfuss2}). We could have also followed this notation
here, but we think that, in this case, it is clearer to keep the
explicit dependence on time of the attractor, which, in addition,
allows us to compare in a more straightforward manner with the
concept of attractor in an autonomous framework.
\end{remark}
From the concept of non-autonomous attractor, we can now give the
following definition of permanence, which will be suitable for
non-autonomous partial differential equations.
\begin{definition}\label{permanence} \rm
We say that a system has the property of pullback
permanence (or that it is permanent in the pullback sense) if there exists a
time-dependent family of sets $U:\mathbb{R}\longmapsto X$, satisfying
\begin{enumerate}
\item $U(t)$ is absorbing for every bounded set $D\subset X$ (cf. Definition
\ref{defi2}).
\item $\mathop{\rm Dist}(U(t),\{0\})>0$ for all $t\in \mathbb{R}$.
\end{enumerate}
\end{definition}
\begin{remark} \rm
This same concept has been also applied to systems of PDEs in
Langa et al. \cite{LRS2}.
\end{remark}
\section{Order-preserving non-autonomous differential
equations}
We now introduce order-preserving systems and the concepts of sub, super and
complete trajectories as a generalization of sub, super and equilibria in
Hess \cite{Hess}, and of sub, super and equilibria in Arnold and Chueshov
\cite{ArnoldChu} in the stochastic case and Chueshov \cite{Chues} in the
non-autonomous case under stronger conditions.
\begin{definition}\rm
The process \{$S(t,s):X\to X\}_{t\geq s}$ is
order-preserving if there exists an order relation `$\leq $' in $X$ such
that, if $u_{0}\leq v_{0}$, then $S(t,s)u_{0}\leq S(t,s)v_{0}$, for all $%
t\geq s$.
\end{definition}
\begin{definition} \rm
Let $S$ be an order-preserving process. We call $\underline{u}$ ($\overline{u%
}):\mathbb{R}\to X$ a sub-trajectory (super-trajectory) of $S$ if it
satisfies
\begin{equation*}
S(t,s)\underline{u}(s)\geq \underline{u}(t),\text{ for all }t\geq s\text{ \
\ (}S(t,s)\overline{u}(s)\leq \overline{u}(t),\text{ for all }t\geq s\text{
).}
\end{equation*}
\end{definition}
\begin{definition} \rm
We call the continuous map $v:\mathbb{R}\to X$ a complete trajectory
if, for all $s\in \mathbb{R}$, we have
\begin{equation*}
S(t,s)v(s)=v(t),\text{ for }t\geq s.
\end{equation*}
\end{definition}
From a sub and super-trajectory $(\underline{u},\overline{u})$ of
a process
such that $\underline{u}(t)\leq \overline{u}(t)$, for all $t\in \mathbb{R}%
,\, $we can define the ``interval''
\begin{equation*}
I_{\underline{u}}^{\overline{u}}(t)=\{u\in X:\ \underline{u}(t)\leq u\leq
\overline{u}(t)\}.
\end{equation*}
Clearly, it is a closed forward invariant set, i.e. $S(t,s)I_{\underline{u}%
}^{\overline{u}}(s)\subset I_{\underline{u}}^{\overline{u}}(t)$, for all $%
t\geq s$.
The following result gives sufficient conditions for the existence of upper
and lower asymptotically stable complete trajectories, giving some
information on the structure of the non-autonomous attractor, adapting to
our case the main results in Arnold and Chueshov \cite{ArnoldChu} and
Chueshov \cite{Chues}. Note that we slightly generalize the results in
\cite{Chues} as we do not impose the set of parameters to be a compact set.
Suppose the pullback attractor attracts time-dependent families of sets in a
base of attraction $\mathcal{D}$.
\begin{theorem} \label{order}
Let $S$ be an order-preserving process and $\mathcal{A}(t)$ its
associated pullback attractor. Let $\underline{u}$, $\overline{u}$ be sub
and super-trajectories such that $\underline{u}(t)\leq \overline{u}(t)$, for
all $t\in \mathbb{R}$, and $I_{\underline{u}}^{\overline{u}}(t)$ the
corresponding associated interval, such that
$\mathcal{A}(t)\subset I_{\underline{u}}^{\overline{u}}(t)$,
for all $t\in \mathbb{R}$ and $\underline{u}$, $\overline{u}\in \mathcal{D}$.
Suppose that there exists $t_0>0$ such that
$S(t_0 +s,s)I_{\underline{u}}^{\overline{u}}(s)$ is relatively
compact, for all $s\in \mathbb{R}$. Then, there exist complete trajectories
$u_{\ast }(t)$, $u^{\ast }(t)\in \mathcal{A}(t)$ such that
\begin{description}
\item[i)]
$\underline{u}(t)\leq u_{\ast }(t)\leq u^{\ast }(t)\leq \overline{u}(t)$,
and $\mathcal{A}(t)\subset I_{u_{\ast }}^{u^{\ast }}(t)$, for all
$t\in\mathbb{R}$.
\item[ii)] $u_{\ast }$ ($u^{\ast })$ is minimal (maximal) in the sense that
it does not exist any complete trajectory in the interval $I_{\underline{u}%
}^{u_{\ast }}(I_{u^{\ast }}^{\overline{u}})$.
\item[iii)] $u_{\ast }(t)$ is globally asymptotically stable from below,
that is, for all $v\in \mathcal{D}$ with $\underline{u}(t)\leq v(t)\leq
u_{\ast }(t)$, for all $t\in \mathbb{R}$, we have that
\begin{equation*}
\lim_{s\to +\infty }d(S(t,-s)v(-s),u_{\ast }(t))=0.
\end{equation*}
$u^{\ast }(t)$ is globally asymptotically stable from above, that is, for
all $v\in \mathcal{D}$ with $u^{\ast }(t)\leq v(t)\leq \overline{u}(t)$, for
all $t\in \mathbb{R}$, we have that
\begin{equation*}
\lim_{s\to +\infty }d(S(t,-s)v(-s),u^{\ast }(t))=0.
\end{equation*}
\end{description}
\end{theorem}
\paragraph{Proof.}
Write $a_{n}(t)=S(t,-nt_{0})\underline{u}(-nt_{0})$,
$b_{n}(t)=S(t,-nt_{0})\overline{u}(-nt_{0})$. Then, we have
\begin{equation}
\underline{u}(t)\leq a_{n}(t)\leq a_{m}(t)\leq b_{m}(t)\leq b_{n}(t)\leq
\overline{u}(t),\text{ \ for \ all }m>n. \label{a22}
\end{equation}
Indeed, $a_{n}(t)=S(t,-nt_{0})\underline{u}(-nt_{0})\geq \underline{u}(t)$,
since $\underline{u}$ is a sub-trajectory. Moreover, for $s=\sigma +r$,
$r>0$,
\begin{eqnarray*}
a_{s}(t) &=&S(t,-st_{0})\underline{u}(-st_{0})=S(t,-(\sigma +r)t_{0})
\underline{u}(-(\sigma +r)t_{0}) \\
&=&S(t,-\sigma t_{0})S(-\sigma t_{0},-(\sigma +r)t_{0})\underline{u}
(-(\sigma +r)t_{0})\geq S(t,-\sigma t_{0})\underline{u}(-\sigma t_{0})\\
&=&a_{\sigma }(t).
\end{eqnarray*}
On the other hand, we have
\begin{eqnarray*}
a_{n+1}(t) &=&S(t,-(n+1)t_{0})\underline{u}(-(n+1)t_{0}) \\
&=&S(t,t-t_{0})S(t-t_{0},-(n+1)t_{0})\underline{u}
(-(n+1)t_{0})\\
&=&S(t,t-t_{0})a_{n+1}(t-t_{0}),
\end{eqnarray*}
and so $a_{n+1}(t)\in S(t,t-t_{0})I_{\underline{u}}^{\overline{u}}(t-t_{0})$,
for all $n\in \mathbb{N}$. Thus, from (\ref{a22}) and the relative
compactness of $S(t,t-t_{0})I_{\underline{u}}^{\overline{u}}(t-t_{0})$,
there exists the following limit
\begin{equation*}
\lim_{n\to +\infty }a_{n}(t)\circeq u_{\ast }(t).
\end{equation*}
Clearly, $u_{\ast }:\mathbb{R}\to X$ is a complete trajectory, as,
by the continuity of the process $S(t,s),$%
\begin{eqnarray*}
S(t,s)u_{\ast }(s) &=&S(t,s)\lim_{n\to +\infty }S(s,-nt_{0})
\underline{u}(-nt_{0}) \\
&=&\lim_{n\to +\infty }S(t,s)S(s,-nt_{0})\underline{u}(-nt_{0})\\
&=&\lim_{n\to +\infty }S(t,-nt_{0})\underline{u}
(-nt_{0})=u_{\ast }(t).
\end{eqnarray*}
We now prove that $u_{\ast }(t)$, $u^{\ast }(t)\in \mathcal{A}(t)$. Indeed,
\begin{equation*}
\mathop{\rm dist}(S(t,s)u_{\ast }(s),\mathcal{A}(t))\leq d(S(t,s)u_{\ast }(s),S(t,s)%
\underline{u}(s))+\mathop{\rm dist}(S(t,s)\underline{u}(s),\mathcal{A}(t)),
\end{equation*}
and the right hand side of the inequality tends to zero when
$s\to
-\infty $. As $S(t,s)u_{\ast }(s)=u_{\ast }(t)$, for all $s\in \mathbb{R}$,
$u_{\ast }(t)\in \mathcal{A}(t)$.
Is is also straightforward to show that, for all $u(t)\in \mathcal{A}(t)$,
\begin{equation*}
\underline{u}(t)\leq u_{\ast }(t)\leq u(t)\leq u^{\ast }(t)\leq \overline{u}
(t),
\end{equation*}
by the definition of $u_{\ast }$ and $u^{\ast }$, the invariance of
$\mathcal{A}(t)$ and the order in $X$.
On the other hand, for any complete trajectory $v(\cdot )$ such that
$\underline{u}(t)\leq v(t)\leq u_{\ast }(t)$, for all $t\in \mathbb{R}$, and
by the order in the process,
\begin{equation*}
u_{\ast }(t)=\lim_{n\to +\infty }S(t,-nt_{0})\underline{u}%
(-nt_{0})\leq \lim_{n\to +\infty }S(t,-nt_{0})v(-nt_{0})=v(t)\leq
u_{\ast }(t),
\end{equation*}
so that $v(t)=u_{\ast }(t)$, for all $t\in \mathbb{R}$. Note that this
implies that $u_{\ast }$ and $u^{\ast }$ are uniquely defined by the order
in $X$.
Finally, for iii), let be $v\in \mathcal{D}$ with $\underline{u}(t)\leq
v(t)\leq u_{\ast }(t)$, for all $t\in \mathbb{R}$. Then, by the attraction
property of $\mathcal{A}(t)$%
\begin{eqnarray*}
u_{\ast }(t)=\lim_{s\to +\infty }S(t,-s)\underline{u}(-s)
&\leq&\lim_{s\to +\infty }S(t,-s)v(-s)\\
&\leq& \lim_{s\to +\infty}S(t,-s)u_{\ast }(-s)=u_{\ast }(t).
\end{eqnarray*}
All these arguments also hold for $u^{\ast }$.
\hfill$\square$
Note that the same conclusions can be got under weaker hypotheses:
\begin{corollary} \label{coro}
Let $S$ be an order-preserving process and $\mathcal{A}(t)$ its
associated pullback attractor. Let $\underline{u}$, $\overline{u}\in
\mathcal{D}$ be such that $\underline{u}(t)\leq \overline{u}(t)$, for all
$t\in \mathbb{R}$, and assume that
\begin{equation*}
\mathcal{A}(t)\subset I_{\underline{u}}^{\overline{u}}(t), \; \forall
t\in \mathbb{R}\text{.}
\end{equation*}
Then there exists two trajectories $u_{\ast }(t)$,
$u^{\ast }(t)\in \mathcal{A}(t)$ such that
\begin{description}
\item[i)] $u_{\ast }(t)\leq u\leq u^{\ast }(t)$, $\forall t\in \mathbb{R}$
and $\forall u\in \mathcal{A}(t)$.
\item[ii)] $u_{\ast }$ ($u^{\ast })$ is minimal (maximal) in the sense that
it does not exist any complete trajectory in the interval $I_{\underline{u}%
}^{u_{\ast }}(I_{u^{\ast }}^{\overline{u}})$.
\item[iii)] $u_{\ast }(t)$ is globally asymptotically stable from below and$%
\ u^{\ast }(t)$ is globally asymptotically stable from above.
\end{description}
\end{corollary}
\paragraph{Proof.}
Since $I_{\underline{u}}^{\overline{u}}(t)\subset \mathcal{D}$, the
attractivity property of $\mathcal{A}(t)$ implies that
\begin{equation*}
\mathop{\rm dist}(S(t,-s)I_{\underline{u}}^{\overline{u}}(-s),\mathcal{A}(t))\to
0,\text{ as }s\to +\infty .
\end{equation*}
Now, the compactness of $\mathcal{A}(t)$ and the order relation in $X$ imply
that there exist $u_{\ast }(t)$, $u^{\ast }(t)$ $\in \mathcal{A}(t)$ with
\begin{equation*}
\lim_{s\to +\infty }S(t,-s)\underline{u}(-s)=u_{\ast }(t)\text{ and }%
\lim_{s\to +\infty }S(t,-s)\overline{u}(-s)=u^{\ast }(t)
\end{equation*}
and the argument follows as in the previous theorem.
\hfill$\square$
\section{Non-autonomous logistic equation}
Let $\Omega $ be a bounded domain in $\mathbb{R}^{N},N\geq 1$, with
smooth boundary $\partial \Omega $. Consider the non-autonomous
logistic equation
\begin{equation}
\begin{gathered}
u_{t}-\Delta u=\lambda u-b(t)u^{3} \\
u\big|_{\partial \Omega }=0, \quad u(s)=u_{0},
\end{gathered} \label{logistic}
\end{equation}
with $\lambda \in \mathbb{R}$ and $b\in C^{\nu }(\mathbb{R})$,
$\nu \in (0,1) $ assuming that there exists a positive constant
$B$ such that
\begin{equation}
\quad 0**0 \mbox{ in } \quad \Omega,
\frac{\partial v}{\partial n}<0 \mbox{ on } \partial \Omega\big\},
$$
where $n$ is the outward unit normal on $\partial \Omega $.
Given a regular domain $D\subset \mathbb{R}^{N}$, $\lambda
_{1}^{D}$ and $\varphi _{1}^{D}$ stand for the principal
eigenvalue and the positive eigenfunction associated to $-\Delta $
under homogeneous Dirichlet condition, normalized such that
$\max_{x\in \bar{D}}\varphi _{1}^{D}(x)=1$. We write $\lambda
_{1}=\lambda _{1}^{\Omega }$ and $\varphi _{1}=\varphi
_{1}^{\Omega }$.
\begin{theorem} \label{existencia}
Assume (\ref{Hb}) and $u_{0}\in \mathcal{V}_{+},{u}_{0}{\neq 0}$.
Then, there exists a unique solution $u(t)=u(t,s;u_{0})\in X$ of
(\ref{logistic}), which is strictly positive for $t>s$.
\end{theorem}
\paragraph{Proof.}
We use the sub-supersolution method, see for instance \cite{Pao}. We take a
domain $D$ such that $\Omega \subset D$ and consider the pair
\begin{equation*}
(\underline{u},\overline{u}):=(0,\varepsilon e^{\gamma (t-s)}\varphi
_{1}^{D}),
\end{equation*}
where $\varepsilon $ and $\gamma $ are constants to be chosen. The pair $(%
\underline{u},\overline{u})$ is a sub-supersolution of $(\ref{logistic})$
provided that
\begin{equation}
0<\frac{\max_{\overline{\Omega }}u_{0}}{\min_{\overline{\Omega }}\varphi
_{1}^{D}}\leq \varepsilon , \label{siete}
\end{equation}
and
\begin{equation}
0\leq \gamma +\lambda _{1}^{D}-\lambda +b(t)\varepsilon ^{2}e^{2\gamma
(t-s)}(\varphi _{1}^{D})^{2}. \label{ocho}
\end{equation}
Now, it is clear that (\ref{siete}) and (\ref{ocho}) are satisfied if $%
\varepsilon $ is large enough and
\begin{equation}
0\leq \gamma +\lambda _{1}^{D}-\lambda \label{cotaa}
\end{equation}
This shows the existence of a nonnegative and nontrivial solution $u$ of $(%
\ref{logistic})$ such that
\begin{equation}
\underline{u}\leq u\leq \overline{u}. \label{estrella}
\end{equation}
Now, the strong maximum principle implies that $u$ is strictly positive for $%
t>s$.
This completes the existence part. The uniqueness follows by a standard way
(Pao \cite{Pao}, Chapter 2).
\hfill$\square$
So, we can define the following flow in $X$, for $t,s\in \mathbb{R}$ and $t\geq
s$, we define $S(t,s):X\to X$ as
$$
S(t,s)u_{0} =u(t,s;u_{0}),
$$
with $u(t,s;u_{0})$ the unique solution of (\ref{logistic}). Furthermore,
(\ref{logistic}) can be written as the following differential equation in $X$:
\begin{equation}
\begin{gathered}
\frac{du(t)}{dt}+Au=\lambda u(t)-b(t)u^{3}(t) \\
u_{|\partial \Omega }=0 \\
u(s)=u_{0}
\end{gathered} \label{alogistic}
\end{equation}
with $A=-\Delta $, the linear operator $A:D(A)\to X$
associated to the Laplacian. Moreover, it is clear that $S(t,s)$
is an order-preserving system. Indeed, it is enough to consider
two initial data $u_0,v_0$, with $u_0\leq v_0$, and apply the
maximum principle to $S(t,s)u_0-S(t,s)v_0$.
\begin{remark} \rm
Note that $v:\mathbb{R}\to X$ is a complete trajectory of problem (%
\ref{logistic}) if
\begin{equation*}
u(t,s;v(s))=v(t)\text{ in }X,\text{ for }t\geq s,\text{ }
\end{equation*}
with $u(t,s;v(s))$ the unique solution of $(\ref{logistic})$ with initial
condition $u(s)=v(s)_{.}$
\end{remark}
\subsection{Asymptotic behaviour forward in time}
We are interested in the study of qualitative properties in the
asymptotic behaviour of problem (\ref{logistic}) when the parameter $\lambda
$ changes. The family of maps $\{S(t,s)\}_{t\geq s}$ will allow us to treat
this problem from a dynamical system point of view.
If we fix the initial time $s$, and for $\lambda <\lambda _{1}$, note that
the asymptotic behaviour of (\ref{logistic}) is determined around the zero
solution, that is, $\{0\}$ is globally asymptotically stable. The following
result shows this fact as an easy consequence of Theorem \ref{existencia}.
\begin{corollary} \label{facil}
Assume (\ref{Hb}) and $\lambda <\lambda _{1}$. Then,
\begin{equation*}
\left| u(t,s;u_{0})\right| _{0}\to 0\quad \mbox{as } t\to +\infty.
\end{equation*}
\end{corollary}
\paragraph{Proof.}
From the monotonicity and continuity of the principal eigenvalue
with
respect to the domain, there exists a domain $D\supset \Omega $ such that $%
\lambda <\lambda _{1}^{D}<\lambda _{1}$. So, according to (\ref{cotaa}) we
can take $\gamma <0$ in Theorem \ref{existencia}. So, by (\ref{estrella})
\begin{equation}
0**__\lambda _{1}$, $\Psi _{\lambda }$ goes to $\infty $ as $%
t\to +\infty $ and $\Theta _{\lbrack \lambda ,B]}$ goes to $\theta
_{\lbrack \lambda ,B]}$, where $\theta _{\lbrack \lambda ,B]}$ is the unique
positive solution of
\begin{equation}
\begin{gathered}
-\Delta u=\lambda u-Bu^{3} \quad \text{in }\Omega \\
u=0 \quad \text{on }\partial \Omega .
\end{gathered} \label{cuatro}
\end{equation}
Hence, when $\lambda >\lambda _{1}$ there exist $V\in C(\overline{\Omega })$
and $t_{0}(u_{0})\in \mathbb{R}$ such that
\begin{equation}
0\lambda _{1}$ the behaviour of the positive
solution of $(\ref{logistic})$ changes drastically. In particular, the
system is not permanent. The next result shows this fact.
\begin{lemma}
Consider (\ref{logistic}) with $\lambda >\lambda _{1}$, ${u}_{0}\in \mathcal{%
V}_{+},{u}_{0}{\neq 0}$ and $\lim_{t\to +\infty }b(t)=~0$. Then,
for all $M>0$ and $t_{0}\in \mathbb{R}$, there exists $t>t_{0}$
such that $\ |u(t,s;u_{0})|_{0}>M$.
\end{lemma}
\paragraph{Proof.}
We argue by contradiction. Assume that there exist a positive constant
$0\lambda _{1}$, we can take $\varepsilon >0$ such that
$\lambda >\lambda _{1}+\varepsilon K^{2}$.
For this $\varepsilon >0$, there exists $t_{1}>0$ such that $b(t)\leq
\varepsilon $ for $t\geq t_{1}$. We define
\begin{equation*}
q(t)=\int_{\Omega }u(t,s;u_{0})\varphi _{1}(x)dx.
\end{equation*}
Multiplying the equation for $u$ by $\varphi _{1}$, integrating
over $\Omega $ and using the Green's formula, we obtain
\begin{equation*}
q'(t)=(\lambda -\lambda _{1}-\varepsilon K^{2})q(t)+\int_{\Omega
}(\varepsilon K^{2}-b(t)u^{2})u(t,s;u_{0})\varphi _{1}(x)dx,
\end{equation*}
and so, by (\ref{*1000}), we get for $t\geq \max \{t_{0},t_{1}\}$
\begin{equation*}
q'(t)\geq (\lambda -\lambda _{1}-\varepsilon K^{2})q(t)\quad
\mbox{and}\quad q(s)>0.
\end{equation*}
This is a contradiction to (\ref{*1000}).
\hfill$\square$
The preceding result implies that there does not exist any bounded
absorbing set for (\ref{logistic}) in the sense of a bounded set
$B\subset X$ such that, for any $D\subset X$ bounded, $\
S(t,s)D\subset B$, for $t$ big enough (Chepyzhov and Vishik
\cite{ChepVishik93}, Temam \cite{Temam}). Thus, at the parameter
value $\lambda =\lambda _{1}$ it occurs a qualitative change of
the asymptotic behaviour of the equation, as a ``disappearance''
of the dissipative effect in the equation. On the other hand, note
that the presence of the term -$b(t)u^{3}$ is also causing some
dissipativity in the problem. It is the possibility of being
$b(t)$ as close to zero as time goes to $\infty $ which causes so
big change in the asymptotic behaviour.
Recently, the theory of global attractors for general non-autonomous
differential equations has been introduced (see Section 2). In the following
section we apply this theory to our problem. Some new qualitative properties
in the asymptotic behaviour of (\ref{logistic}) will arise by using this
theory. In particular, we will show a result on pullback permanence.
\subsection{Existence of non-autonomous attractors for the logistic equation}
In this Section we will prove the existence of a compact absorbing
set in $X$. In fact, we will prove the existence of a compact
absorbing set in $C_{0}^{1}(\bar{\Omega})$ by the existence of a
bounded absorbing ball in $C_{0}^{2}(\bar{\Omega})$. We will do it
in two steps:
\paragraph{Absorbing set in $X$.}
Consider the non-autonomous differential equation
\begin{gather*}
\frac{dy(y)}{dt}=\lambda y(t)-b(t)y^{3}(y) \\
y(s)=y_{s}
\end{gather*}
whose solution satisfies
\begin{equation*}
y^{2}(t,s;y_{s})=\frac{e^{2\lambda t}}{\frac{e^{2\lambda s}}{y_{s}^{2}}%
+2\int_{s}^{t}e^{2\lambda \tau }b(\tau )d\tau }.
\end{equation*}
Now, given $D\subset X$ bounded, i.e., $\sup_{d\in D}|d|\leq M$, for $M>0$,
and $u_{0}\in D$, the pair $(0,y(t,s;M))$ is a sub-supersolution of (\ref
{logistic}) and so,
\begin{equation*}
u(t,s;u_{0})\leq y(t,s;M),\text{ \ for all }t\geq s \: \,
\mbox{and all} \, \: u_0\in D.
\end{equation*}
Thus, there exists $T(t)\in \mathbb{R}$ such that
\begin{equation}
\left| u(t,s;u_{0})\right| _{0}\leq r_{1}(t)\text{ \ \ \ for \ \ }s\leq T(t)
\label{r1}
\end{equation}
where
\begin{equation*}
r_{1}^{2}(t)=\frac{e^{2\lambda t}}{\int_{-\infty }^{t}e^{2\lambda \tau
}b(\tau )d\tau }.
\end{equation*}
Clearly, this means that the ball in $X$ with radius $r_{1}(t)$, $%
B_{X}(0,r_{1}(t))$, is absorbing for the process $\ S(t,s)$.
\paragraph{Absorbing set in $C_{0}^{1}(\bar{\Omega})$.}
To obtain a family of absorbing sets in $C_{0}^{1}(\bar{\Omega})$
we need the following result which follows by \cite{Mora}, see also Lemma
3.1 in \cite{CantrellCosnerHu}. Here, for a Banach space $Y$, $Y^{\beta }$
will denote the usual fractional power spaces with norm $\left| \cdot
\right| _{\beta }$.
\begin{lemma} \label{lemamora}
The operator $A$ generates an analytic semigroup on
$Y=C_{0}^{k}(\bar{\Omega})$ for $k=0,1$. Moreover, it holds
\begin{equation*}
Y^{\beta }\hookrightarrow C_{0}^{k+q}(\bar{\Omega})\text{ for }q=0,1\text{
and }2\beta >q.
\end{equation*}
\end{lemma}
Given $D\subset X$ bounded, i.e., $\sup_{d\in D}|d|\leq M$, for $M>0$, take $%
u_{0}\in D$. We define
\begin{equation*}
h(r,s)=\lambda u(r,s;u_{0})-b(r)u^{3}(r,s;u_{0})\text{ for }r\geq s.
\end{equation*}
Then, writing the equation from the variation of constants formula, we obtain
\begin{equation*}
u(t,s;u_{0})=e^{-A(t-s)}u_{0}+\int_{s}^{t}e^{-A(t-r)}h(r,s)dr.
\end{equation*}
Hence, taking it between $t-1$ and $t$, we get, for $s\leq t-1$,
\begin{equation*}
u(t,s;u_{0})=e^{-A}u(t-1,s;u_{0})+\int_{t-1}^{t}e^{-A(t-r)}h(r,s)dr.
\end{equation*}
Hence,
\begin{eqnarray*}
\left| u(t,s;u_{0})\right| _{\beta }
&=&\left| A^{\beta }u(t,s;u_{0})\right|_{0}\\
&\leq& \left\| A^{\beta }e^{-A}\right\| _{op}\left| u(t-1,s;u_{0})\right|_{0}\\
&&+\sup_{r\in \lbrack t-1,t]}\left| h(r,s)\right| _{0}\int_{t-1}^{t}\big\|
A^{\beta }e^{-A(t-r)}\big\| _{op}dr.
\end{eqnarray*}
Now, using the estimate $\left\| A^{\beta }e^{-A(t-r)}\right\| _{op}\leq
C_{\beta }(t-r)^{-\beta }e^{-\delta (t-r)}$ for some constants $C_{\beta
},\delta >0$ (cf. Henry \cite{Henry}) and (\ref{r1}), we obtain the
existence of $M(t)$ and $T_{0}(t)$ such that
\begin{equation*}
\left| u(t,s;u_{0})\right| _{\beta }\leq M(t)\text{ for all }s\leq T_{0}(t)
\end{equation*}
with $\beta <1-\varepsilon $, and any $\varepsilon \in (0,1)$. Applying now
Lemma \ref{lemamora} with $q=1$ and $\beta >1/2$, we obtain
\begin{equation*}
\left| u(t,s;u_{0})\right| _{C^{1}}\leq R_{1}(D,t)\text{ for all }s\leq
T_{0}(t),
\end{equation*}
and then the ball in $C_{0}^{1}(\bar{\Omega})$, $B(0,R_{1}(t))$ is
absorbing in $C_{0}^{1}(\bar{\Omega})$.
We can repeat the argument taking now $Y=C_{0}^{1}(\bar{\Omega})$ and $D$ a
bounded set in $Y$. In this case, using again Lemma \ref{lemamora}, we
obtain that
\begin{equation*}
\left| u(t,s;u_{0})\right| _{C^{2}}\leq R_{2}(D,t)\text{ for all }s\leq
T_{1}(t),
\end{equation*}
and hence, the existence of an absorbing set in
$C_{0}^{2}(\bar{\Omega})$, and so compact in $X$ or $Y$.
\begin{remark} \label{18} \rm
From (\ref{r1}) we conclude that the non-autonomous attractor $%
\mathcal{A}(t)$ attracts not only the ``pullback pseudotrajectories''
$\cup_{s\leq t}S(t,s)u_{0}$, but we have a stronger attraction property:
Consider the \textit{base of attraction}
\begin{equation*}
\mathcal{D}=\{v:\mathbb{R}\to X\text{ continuous, such that, }%
\lim_{s\to -\infty }\frac{e^{2\lambda s}}{\left| v(s)\right| _{0}^{2}%
}=0\},
\end{equation*}
that is, $\mathcal{D}$ is the set of \textit{tempered} functions,
which is also usually defined in the literature as the
\textit{base of attraction} (see, for example, Schmalfuss
\cite{Schmalfuss2} or Flandoli and Schmalfuss \cite{Fl-Sch}).
Then, we have that, given $v\in \mathcal{D},$%
\begin{equation}
\lim_{s\to -\infty }\mathop{\rm dist}(S(t,s)v(s),\mathcal{A}(t))=0. \label{27}
\end{equation}
Indeed, we have that for $s$ small enough
\begin{equation*}
|u(t,s;v(s))|_{0}^{2}\leq \frac{e^{2\lambda t}}{\frac{e^{2\lambda s}}{\left|
v(s)\right| _{s}^{2}}+2\int_{s}^{t}e^{2\lambda \tau }b(\tau )d\tau }\leq
r_{1}^{2}(t).
\end{equation*}
Note that every map $v$, with $v(t)\equiv v_{0}$, for all $t$, is in $%
\mathcal{D}$.
\end{remark}
\subsection{Upper semicontinuity of non-autonomous attractors to the global
attractor}
Let $b^{\sigma }$ be a family of functions satisfying (\ref{Hb}) and $S_{\sigma
}(t,s)$ be the non-au\-tonomous dynamical system associated to
\begin{equation}
u_{t}-\Delta u=\lambda u-b^{\sigma }(t)u^{3},\quad \lim_{\sigma
\searrow 0}b^{\sigma }(t)=\alpha >0 \label{asigmalogistic}
\end{equation}
uniformly on bounded sets of $t\in \mathbb{R}$, $\lambda >\lambda _{1}$,
defined as a small perturbation of the given semigroup $S_{0}$ associated to
the autonomous equation
\begin{equation}
u_{t}-\Delta u=\lambda u-\alpha u^{3}. \label{autonlogistic}
\end{equation}
\begin{remark} \rm
Note that this holds, for example, for $0__**s$,
\begin{equation}
\lim_{\sigma \searrow 0}d(S_{\sigma }(t,s)u_{0},S_{0}(t)u_{0})=0 \label{h1}
\end{equation}
uniformly on bounded sets of $X$.
On the other hand, suppose that there exist the pullback attractors
${\cal A}_{\sigma }(t)$ and ${\cal A}$, associated with $S_{\sigma }$
and $S_{0}$ respectively, such that
${\cal A}_{\sigma }(t)\subset K_{\sigma }(t)$,
${\cal A}\subset K$, where $K_{\sigma }(t)$ and $K$ are compact absorbing sets
associated to the corresponding flows, and satisfying
\begin{equation}
\lim_{\sigma \searrow 0}\mathop{\rm dist}(K_{\sigma }(t),K)=0,\text{ \ for every \ }t\in
\mathbb{R}. \label{h2}
\end{equation}
Then we have (Caraballo and Langa \cite{CaraballoLa})
\begin{theorem} \label{22}
Under the preceding assumptions (\ref{h1}), (\ref{h2}), it follows \ that,
for all \ $t\in \mathbb{R}$,
\begin{equation*}
\lim_{\sigma \searrow 0}\mathop{\rm dist}({\cal A}_{\sigma }(t),
{\cal A})=0.
\end{equation*}
\end{theorem}
It remains to prove that conditions (\ref{h1}) and (\ref{h2}) are satisfied in
our case. Indeed, (\ref{h2})\ is a consequence of the expression for $%
r_{1}^{\sigma }(t)$, with $r_{1}^{\sigma }(t)$ the corresponding
radius of the absorbing ball associated to (\ref{logistic}) with
$b^{\sigma }(t)$.
From (\ref{r1}),
\begin{equation*}
\lim_{\sigma \searrow 0}r_{1}^{\sigma }(t)^{2}=\lim_{\sigma \searrow 0}\frac{%
e^{2\lambda t}}{\int_{-\infty }^{t}e^{2\lambda \tau }b^{\sigma }(\tau )d\tau
}=2\frac{\lambda }{\alpha },
\end{equation*}
which is independent of $t\in \mathbb{R}$, so that the same is true for
$R_{2}^{\sigma }(t)$ and (\ref{h2}) holds.
On the other hand, (\ref{h1})\ is the content of the following lemma
\begin{lemma}
Given $u_{\sigma }(t,s;u_{0})$, $u(t;u_{0})$ solutions of
(\ref{asigmalogistic}) and (\ref{autonlogistic}) respectively with initial data
$u_{\sigma }(s)=u(s)=u_{0}$, it holds that, for all $t>s$,
\begin{equation}
\lim_{\sigma \searrow 0}|u_{\sigma }(t,s;u_{0})-u(t;u_{0})|_{0}=0.
\label{a11}
\end{equation}
\end{lemma}
\paragraph{Proof.}
Since $\lim_{\sigma \searrow 0}b^{\sigma }(t)=\alpha >0$, for $\sigma $
sufficiently small, positive constants are supersolutions of
(\ref{asigmalogistic}), see (\ref{supercons}), and so
\begin{equation}
\left| u_{\sigma }(t,s;u_{0})\right| _{0}\leq M\text{ (independent of }
\sigma ). \label{inde}
\end{equation}
Calling $v_{\sigma }(t,s;u_{0})=u_{\sigma }(t,s;u_{0})-u(t;u_{0})$ and
using the variation of constants formula, we have
\begin{eqnarray*}
v_{\sigma }(t,s;u_{0}) &=&\int_{s}^{t}e^{-A(t-r)}(\lambda v_{\sigma
}(r,s;u_{0})+(\alpha -b^{\sigma }(r))u_{\sigma }^{3}(r,s;u_{0})\\
&&+\alpha (u_{\sigma }^{3}(r,s;u_{0})-u^{3}(r;u_{0})))dr.
\end{eqnarray*}
Since $\left\| e^{-A(t-r)}\right\| _{op}\leq e^{-\delta (t-r)}\leq 1$, we
get
\begin{eqnarray*}
\left| v_{\sigma }(t,s;u_{0})\right| _{0} &\leq &\lambda \int_{s}^{t}\left|
v_{\sigma }(r,s;u_{0})\right| _{0}dr+\int_{s}^{t}\left| \alpha -b^{\sigma
}(r)\right| \sup_{\sigma }\left| u_{\sigma }^{3}(r,s;u_{0})\right| _{0}dr \\
&&+3\alpha \int_{s}^{t}\eta _{\sigma }^{2}(\xi _{r})\left| v_{\sigma
}(r,s;u_{0})\right| _{0}dr.
\end{eqnarray*}
Using now (\ref{inde}) and Gronwall's lemma, we get (\ref{a11}).
\hfill$\square$
\subsection{Bifurcation scenario for positive solutions}
In this section we describe the changes in the asymptotic
behaviour of the equation as the parameter value crosses $\lambda
_{1}$. For $\lambda <\lambda _{1}$, note that the zero solution is
globally asymptotically stable, in both the forward and the
pullback sense. Indeed, from (\ref{ceroo}) we have that
\begin{equation*}
\lim_{t\to +\infty }|u(t,s;u_{0})|_{0}^{2}=\lim_{s\to
-\infty }|u(t,s;u_{0})|_{0}^{2}=0.
\end{equation*}
This means that, in this case, the non-autonomous attractor reduces to a
fixed (not depending on time) point, i.e. $\mathcal{A}(t)\equiv \{0\}$, for
all $t\in \mathbb{R}$.
On the other hand, a nontrivial attractor exists for values of the parameter
bigger than $\lambda _{1}$. In particular, we prove that the attractor is
bigger than the zero solution, i.e. $\{0\}$ $\varsubsetneq \mathcal{A}(t)$.
\begin{proposition}
\label{stricposi}Given $u_{0}\in C_{0}^{1}(\bar{\Omega})$ strictly positive,
$\lambda >\lambda _{1}$ and $t\in \mathbb{R}$, there exists $\varepsilon >0$
such that, for all $s\leq t$%
\begin{equation*}
|S(t,s)u_{0}|_{0}>\varepsilon .
\end{equation*}
\end{proposition}
\paragraph{Proof.}
Since $\lambda >\lambda _{1}$ and $u_{0}\in C_{0}^{1}(\bar{\Omega})$ is
strictly positive, it is not hard to prove that $\underline{u}\mathbb{=}%
\varepsilon \varphi _{1}$ is a subsolution of (\ref{logistic}) provided that
$\varepsilon $ verifies
\begin{equation*}
0<\varepsilon \leq \min_{x\in \bar{\Omega}}\frac{u_{0}(x)}{\varphi
_{1}(x)}\quad\mbox{and}\quad \varepsilon ^{2}B\leq \lambda -\lambda
_{1}.
\end{equation*}
Taking $\varepsilon $ sufficiently small, we get
$\varepsilon \varphi _{1}\leq u(t,s;u_{0})$ whence the result follows.
\hfill$\square$
\begin{remark} \rm
Note that this implies that $\{0\}$ $\varsubsetneq \mathcal{A}(t)$. In fact,
there exists a subset of $\mathcal{A}(t)$ bounded away from zero
``attracting'' every $u_{0}\in C_{0}^{1}(\bar{\Omega})$ strictly positive.
\end{remark}
\paragraph{On the structure of the pullback attractor.}
In this Section we apply the results of Section 3 (Corollary \ref{coro}) to
our equation (\ref{logistic}).
We take $(\underline{u}(t),\overline{u}(t))=(0,r_{1}(t))$. Observe
that $r_{1}(t)\in \mathcal{D}$, because
\begin{equation*}
\lim_{s\to -\infty }\frac{e^{2\lambda s}}{r_{1}^{2}(s)}=0.
\end{equation*}
Moreover, $\mathcal{A}(t)\subset I_{0}^{r_{1}}(t)$, by (\ref{r1}). So,
applying Corollary \ref{coro}, there exists a complete trajectory
$u^{\ast}(\cdot )$ in the pullback attractor. Since
$\mathcal{A}(t)\neq \{0\}$, we have that $u^{\ast }(t)\neq \{0\}$,
for all $t\in \mathbb{R}$.
\begin{remark} \rm
Note that $u^{\ast }(\cdot )\in $ $\mathcal{A}(\cdot )$ is a very special
complete trajectory. On the one hand, it is a maximal trajectory, an upper
bound on the pullback attractor in the positive cone, that is, for all $%
u(t)\in \mathcal{A}(t)$
\begin{equation*}
u(t)\leq u^{\ast }(t).
\end{equation*}
On the other hand, it is globally asymptotically stable from above
(cf. Theorem~\ref{order}).
\end{remark}
Finally, we have the following result.
\begin{theorem}
Assume (\ref{Hb}) and $\lambda >\lambda _{1}$. Then, (\ref{logistic}) is permanent
in a pullback sense.
\end{theorem}
\paragraph{Proof.}
Given a bounded set $D\subset X$ and $u_{0}\in D$, by (\ref{cotaautonomo})
we have that
\begin{equation*}
0**